Arc Length and Radians

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Learning goals

Calculate arc lengths from central angles measured in degrees
Justify why the arc-length to radius ratio depends only on the angle
Interpret radians and convert between degree and radian measures
Find lengths of selected minor and major arcs using radians
Recover radii or angles after expressing lengths in matching units

A fraction of a turn gives a fraction of the circumference

A central turn is the rotation from a starting radius to an ending radius along the selected circular route.

An arc length measures distance along a selected curved part of a circle. It is different from the arc’s degree measure and from the straight chord joining its endpoints.

A $90^\circ$ turn uses one quarter of a circle. On a circle of radius $6$ cm, the full circumference is $12\pi$ cm, so that arc is $3\pi$ cm long. Equal central turns cut equal arcs in the same circle because a rotation about its center matches one arc to the other. Splitting a turn into equal parts therefore splits the circumference into the same number of equal lengths.

The previous lesson established the circumference rule and reasoning with arbitrarily close bounds. For a turn not exactly given by those equal subdivisions, trap its measure between finer fractional turns: their arc lengths trap the selected length, and both gaps can be made as small as desired.

If a circle has positive radius $r$ and the selected arc has central turn $\alpha$ degrees, its length is

s=\frac{\alpha}{360}\cdot2\pi r.

Here $\alpha$ is the numerical degree measure. For a single traversal, it lies from $0$ to $360$ ; zero gives no travel, and $360$ gives the whole circumference.

Worked example 1 Measure part of the circumference

Problem. A circle has radius $7$ cm. Find the length of the selected $72^\circ$ minor arc, first exactly and then to the nearest hundredth of a centimeter.

The blue route is the selected arc. Its chord is not drawn.

Solution. The central turn occupies $72/360$ of a full turn. Apply that fraction to the circumference.

\begin{aligned} s&=\frac{72}{360}\cdot2\pi\cdot7,\\ s&=\frac{14\pi}{5}\text{ cm},\\ s&\approx8.80\text{ cm}. \end{aligned}

Keep the exact expression until the final rounding.

Check your understanding

A circle has radius $10$ cm. Find the length of the selected $126^\circ$ arc.

7\pi

cm B

14\pi

cm C

\tfrac72\pi

cm D

126

The ratio that removes the circle’s size

If you enlarge a circle without changing the central turn, the selected arc and the radius grow by the same factor.

A fixed turn keeps arc length over radius unchanged#

Start with a radius of $6$ cm and a selected arc of $4$ cm. Enlarge the circle by a factor of $5/2$ . The new radius is $15$ cm and the matching arc is $10$ cm.

The blue and dashed red arcs have the same central turn. Each length is multiplied by the same scale factor.

Why does enlargement scale a curved arc? Every chord in a polygonal approximation scales by $5/2$ , so the sum of those chord lengths does too. Making the approximation arbitrarily close to the arc, as in the preceding lesson, preserves that same factor for the curved length. The ratios are

\begin{aligned} \frac{4}{6}&=\frac23,\\ \frac{10}{15}&=\frac23. \end{aligned}

Now let a positive dilation factor $k$ map a circle of radius $r$ and selected arc length $s$ to a larger or smaller circle. Matching the same central turn gives radius $kr$ and arc length $ks$ . Since $r$ and $k$ are positive,

\begin{aligned} \frac{ks}{kr}&=\frac{s}{r}. \end{aligned}

The chosen numerical lengths played no special role. Rotations and translations can first align the centers and starting rays, so the argument applies to matching central turns in any two circles.

Worked example 2 Transfer an arc between circles

Problem. The same central angle cuts an arc of length $14$ cm in a circle of radius $8$ cm. Find the matching arc length in a circle of radius $20$ cm. Then find the arc-length to radius ratio in each circle.

The two selected arcs correspond under a dilation. Use the stated lengths.

Solution. The radius grows by the factor $20/8$ . The matching arc grows by that factor too.

\begin{aligned} s&=14\cdot\frac{20}{8},\\ s&=35\text{ cm}. \end{aligned}

Compare lengths in the same unit before dividing.

\begin{aligned} \frac{14\text{ cm}}{8\text{ cm}}&=\frac74,\\ \frac{35\text{ cm}}{20\text{ cm}}&=\frac74. \end{aligned}

The lengths differ, but the common ratio records the same turn.

Check your understanding

The same central angle cuts arcs in circles of radii $15$ cm and $25$ cm. The first arc is $21$ cm long. How long is the second?

31

cm B

35

cm C

\tfrac{63}{5}

cm D

21

A radian measures how many radii fit along the arc

Take an arc $3$ cm long in a circle of radius $3$ cm. The arc is one radius long. The central turn it selects is one radian.

One radian on a circle of radius 3 cm

The radius r is 3 units. The central angle is 1 radian (about 57.30°). The selected arc length s is 3 units. The ratio s/r is 1. This arc is exactly one radius long.

One radian selects an arc equal in length to the radius, measured along the curve. Lengths in this figure are in centimeters, so both lengths are 3 cm.

More generally, the radian measure of a central turn is its selected arc length divided by the positive radius, using matching length units.

\theta=\frac{s}{r}.

The ratio is independent of circle size by the proof above. It counts radii along the curve, not straight chords. An angle of $2$ radians selects an arc two radii long; it need not involve the number $\pi$ .

The length units cancel in the quotient. The word radian, or the abbreviation $\mathrm{rad}$ , still identifies the result as an angle measure. A radius of $5$ cm is a length; an angle of $5$ radians is a turn. They are different kinds of quantities.

Choose an angle, hold it fixed, and vary the positive radius. Predict how the selected arc length changes and whether its ratio to the radius changes. Then select exactly one radian and compare the two lengths.

Compare arc length with radius

Radius Angle

The radius r is 2 units. The central angle is π/2 radians (90°). The selected arc length s is π units. The ratio s/r is π/2. The arc length and radius use the same length unit.

Keep the angle fixed while changing the positive radius. Compare the selected curved arc with a radius, then choose one radian.

A full turn uses the entire circumference. Dividing its length by its radius gives

\theta_{\text{full}}=2\pi.

So one full turn is $2\pi$ radians and one half-turn is $\pi$ radians.

Convert the unit without changing the turn

The equivalence of a half-turn in the two units gives the conversion factors.

\begin{gathered} 180^\circ=\pi\text{ radians},\\ \theta_{\text{radians}}=\alpha_{\text{degrees}}\cdot\frac{\pi}{180},\\ \alpha_{\text{degrees}}=\theta_{\text{radians}}\cdot\frac{180}{\pi}. \end{gathered}

For example, dividing a half-turn into six equal parts shows that $30^\circ$ is $\pi/6$ radians. You are renaming the turn, not changing it.

Fraction of a full turn	Degrees	Radians
No turn	$0^\circ$	$0$
One eighth	$45^\circ$	$\pi/4$
One quarter	$90^\circ$	$\pi/2$
One half	$180^\circ$	$\pi$
Three quarters	$270^\circ$	$3\pi/2$
One full turn	$360^\circ$	$2\pi$

Worked example 3 Change angle units in either direction

Problem. Convert each angle exactly.

Case A: $135^\circ$ to radians.

Case B: $11\pi/10$ radians to degrees.

Solution. Case A: multiply the numerical degree measure by $\pi/180$ .

135\cdot\frac{\pi}{180}=\frac{3\pi}{4}.

Case B: multiply the radian measure by $180/\pi$ .

\frac{11\pi}{10}\cdot\frac{180}{\pi}=198^\circ.

The first angle is below a half-turn in either unit. The second is above a half-turn in either unit.

Check your understanding

Convert $\tfrac{5\pi}{12}$ radians to degrees.

150^\circ

60^\circ

75^\circ

15^\circ

Use radians to find a selected arc

A radian measure tells you the arc length in units of the radius. Use that meaning before multiplying.

Worked example 4 Follow the major route

Problem. A circle has radius $9$ cm. Its selected major arc from $A$ through $C$ to $B$ has central turn $7\pi/6$ radians. Find that arc’s exact length.

The solid blue route passes through C. The dashed red route is the other arc between A and B.

Solution. The selected arc is $7\pi/6$ radii long. Equivalently, its length divided by $9$ cm is $7\pi/6$ . Multiply the ratio by the radius.

\begin{aligned} s&=9\cdot\frac{7\pi}{6},\\ s&=\frac{21\pi}{2}\text{ cm}. \end{aligned}

The turn is between $\pi$ and $2\pi$ , so its length is between half the circumference and the full circumference, as a major arc should be.

For any positive radius and selected central turn measured in radians, the definition rearranges to

s=r\theta.

The degree formula and this radian formula agree because converting a degree measure multiplies it by $\pi/180$ . The input to $s=r\theta$ must be a radian measure.

Between distinct endpoints, the minor arc uses a turn below $\pi$ radians and the major arc uses the remaining turn above $\pi$ . Their turns total $2\pi$ radians and their lengths total the circumference. Opposite endpoints give two semicircles, each with turn $\pi$ radians.

Check your understanding

A circle has radius $11$ cm. The minor central angle is $\tfrac{2\pi}{11}$ radians. Find the length of the major arc with the same endpoints.

2\pi

cm B

22\pi

cm C

18\pi

cm D

20\pi

Recover a radius or an angle, with units intact

An unknown radius is the arc length divided by the radian measure. An unknown radian measure is the arc length divided by the radius. Both conclusions come from the same defining ratio.

Worked example 5 Recover the circle from an arc

Problem. A selected minor arc is $22$ cm long and its central angle is $4/5$ radian. Find the circle’s radius and diameter.

The stated arc length follows the blue curve. The radius is unknown.

Solution. Divide the arc length by the radian measure to recover the radius.

\begin{aligned} r&=22\div\frac45,\\ r&=\frac{55}{2}\text{ cm}. \end{aligned}

The diameter is twice the radius.

d=55\text{ cm}.

Multiplying the radius by $4/5$ returns the given arc length, so the result fits the circle.

The ratio needs one length unit throughout. For an arc measured in meters and a radius in centimeters, convert one before dividing. Otherwise the bare numerical quotient changes with your choice of units.

A selected arc traversed once cannot have a turn greater than $2\pi$ radians. A zero turn gives zero distance; it is the boundary case, not a positive-length arc between distinct endpoints. A journey with a complete lap followed by an extra turn repeats part of the circle, so add the full-lap distance and the extra arc rather than calling it one simple arc.

Check your understanding

An arc is $0.9$ m long in a circle of radius $36$ cm. What is its central angle in radians?

\tfrac52

\tfrac25

\tfrac1{40}

25

Arc Length and Radians

Learning goals

A fraction of a turn gives a fraction of the circumference

The ratio that removes the circle’s size

A radian measures how many radii fit along the arc

Convert the unit without changing the turn

Use radians to find a selected arc

Recover a radius or an angle, with units intact

Common mistakes

Practice

Multiple Choice Questions (MCQ)

Core practice

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